```Date: Feb 10, 2013 4:38 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Matheology § 214

Matheology § 214What?s wrong with the axiom of choice?Part of our aversion to using the axiom of choice stems from our viewthat it is probably not ?true?. {{In fact it is true for existing sets- but there it is not required as an axiom but is a self-evidenttruth.}} A theorem of Cohen shows that the axiom of choice isindependent of the other axioms of ZF, which means that neither it norits negation can be proved from the other axioms, providing that theseaxioms are consistent. Thus as far as the rest of the standard axiomsare concerned, there is no way to decide whether the axiom of choiceis true or false. This leads us to think that we had better reject theaxiom of choice on account of Murphy?s Law that ?if anything can gowrong, it will?. This is really no more than a personal hunch aboutthe world of sets. We simply don?t believe that there is a functionthat assigns to each non-empty set of real numbers one of itselements. While you can describe a selection function that will workfor ?nite sets, closed sets, open sets, analytic sets, and so on,Cohen?s result implies that there is no hope of describing a de?nitechoice function that will work for ?all? non-empty sets of realnumbers, at least as long as you remain within the world of standardZermelo-Fraenkel set theory. And if you can?t describe such afunction, or even prove that it exists without using some relative ofthe axiom of choice, what makes you so sure there is such a thing?Not that we believe there really are any such things as in?nite sets,or that the Zermelo-Fraenkel axioms for set theory are necessarilyeven consistent. Indeed, we?re somewhat doubtful whether large naturalnumbers (like 80^5000, or even 2^200) exist in any very real sense,and we?re secretly hoping that Nelson will succeed in his program forproving that the usual axioms of arithmetic?and hence also of settheory?are inconsistent. (See E. Nelson. Predicative Arithmetic.Princeton University Press, Princeton, 1986.) All the more reason,then, for us to stick with methods which, because of their concrete,combinatorial nature, are likely to survive the possible collapse ofset theory as we know it today.[Peter G. Doyle, John Horton Conway: "Division by Three" 1994, ARXIVmath/0605779v1]http://arxiv.org/abs/math/0605779v1Regards, WM
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